0
$\begingroup$

I want to find out in what range a certain function is convex. In order to find out, I calculated partial derivatives and set up a Hessian matrix. As the Hessian still has variables in it, I am not sure about the interpretation. I would like to work with the method of principal minors.

I want to make use of this theorem:

Theorem: $f(x,y)$ is convex if and only if its $n \times n$ Hessian matrix is positive semidefinite for all possible values of $(x,y)$. The Hessian is positive definite if and only if its $n=2$ leading principal minors are positive.

My $2\times 2$ matrix: $$\begin{bmatrix} 6x+4 & 7855\\7855 & 2\end{bmatrix}$$

$\endgroup$

1 Answer 1

0
$\begingroup$

The leading principal minors are

  1. $6x+4$
  2. The determinant of the full matrix, $(6x+4)2 - 7855^2$.

For what values of $x$ are both of these positive? Those are the values of $x$ for which your function is (strictly) convex. (Note that in principle you only need to check the determinant, because the bottom-right entry is already positive ($2$) and you can permute the rows and columns so that it becomes the top-right leading principal minor instead of $6x+4$).

In this case there is only a single point where the Hessian is semi-definite, so you don't need to deal with the semi-definite case (you can include the point in the convex set).

$\endgroup$
1
  • $\begingroup$ So this means the Hessian is positive definite (function is strictly convex) if x > -4/6 and the Hessian is positive semidefinite (at x = -4/6)? What can we say about the function at x = -4/6 and can we say anything about concavity also or do we need to make additional calculations in order to say anything about concavity? $\endgroup$
    – user824469
    Sep 18, 2020 at 19:30

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.